There is a gentle and overbearing word in mathematics, which has only what meaning.

Yes, it means existence. There is such a geometric figure.

Unique means unique. Only one, not many.

For example:

(1) A point outside the straight line, with one and only one straight line perpendicular to the known straight line;

It means that beyond the straight line, there must be a straight line perpendicular to the known straight line, and there is only one, and there will be no second one.

(2) At a point other than a straight line, one and only one straight line is parallel to the known straight line.

It means that at a point outside the straight line, there must be a straight line parallel to the known straight line, and there is only one, and there will be no second one.